3. (Minimum-variance Hedging) A farmer has a crop of grapefruit juice that will be ready for harvest and sale as 150000 pounds of grapefruit juice in 3 months. He is worried about possible price changes, so he is considering hedging - a financial engineering technique that minimizes future uncertainties in the cash flow. Typically, hedging is carried out using futures contract. However, unfortunately, there is no futures contract for grapefruit juice, but there is a futures contract for orange juice. Still, the farmer might consider using the futures contract for orange juice as a replacement for futures contract for grapefruit juice, in the hope that these two contracts are highly correlated due to the similarity of the underlying products. Currently, the spot prices are $1.20 per pound for orange juice and $1.50 per pound for grapefruit juice. The standard deviation of the prices of orange juice and grapefruit juice is about 20% per year, and the correlation coefficient between their prices is about 0.7 (highly correlated). What is the minimum variance hedge for farmer, i.e. how many orange juice futures contracts does the farmer need to purchase in total? Hint: let h be the number of orange juice futures contracts that the farmer purchases. Let T be the time when the grapefruit juice will be sold in 3 months). The cash flow at time T will be two parts: (1) profits from selling the juice (2) cash flow generated by futures contract. Can you write down the explicit form of this cash flow? Notice that this cash flow is a random variable, then we can compute h by minimizing the variance of the cash flow (hence minimum-variance hedging). 131250 Hint: We should be able to write the final cash flow in the form of X + cY where X, Y are the payoff coming from sales and futures contract respectively.cis the variable to be determined. We can minimize its variance by computing V[X$ + cY] = V[X] + c2V[Y] + 2cC[X,Y] where C[X, Y] is the covariance between X, Y and V[X],V[Y] are the variance. We can determine c by setting the its derivative to zero